On the Harris recurrence of iterated random Lipschitz functions and related convergence rate results
type: Item
instance of: scholarly article
title: On the Harris recurrence of iterated random Lipschitz functions and related convergence rate results
zbMATH Open document ID: 1022.60067
DOI: 10.1023/A:1022290807360
author: Gerold Alsmeyer
published in: Journal of Theoretical Probability
publication date: 2003-04-27T00:00:00Z
review text: In the introduction the author considers a sequence of the form \(M_n= F(\theta_n,M_{n-1})\), \(n\geq 1\), which satisfies four assumptions. \((M_n)_{n\geq 0}\) defines a temporally homogeneous Markov chain called an iterated function system (IFS) of i.i.d. Lipschitz maps hereafter. Given a Lipschitz map \(f:\chi\to\chi\), define its Lipschitz constant as \(I(f)\overset{\text{def}}= \sup_{x\neq y}{d(f(x), f(y))\over d(x,y)}\). Put \(L_n\overset{\text{def}}= I(F_n)\) for \(n\geq 1\) and note that \((L_n)_{n\geq 1}\) forms a sequence of i.i.d. random variables which is independent of \(M_0\). Its distribution does therefore not depend on the distribution of \(M_0\), that is, it is the same under which P. Elton (1990) showed that, under every \(P_x\), \((M_n)_{n\geq 0}\) converges weakly to a stationary distribution \(\pi\) provided that its Lyapunov exponent \(\ell^*\) is a.s. negative, i.e. \[ \ell^*\overset{\text{def}}= \lim n^{-1}\log\ell(F_n\circ\cdots\circ F_1)< 0\tag{1} \] and \[ \mathbb{E}\log^x L_1<\infty,\quad \mathbb{E}\log^+ d(F_1(x_0),x_0)< \infty\tag{2} \] for some \(x_0\in\chi\). He showed that \(\pi\) is unique and \((M_n)_{n\geq 0}\) is ergodic under \(P_\pi\). By Birkhoff's ergodic theorem, \[ \lim_{n\to\infty} {1\over n} \sum^n_{k=1} \mathbf{1}_B(M_k)= \pi(B)\quad P_\pi\text{-a.s.}; \] and thus also \(P_x\)-a.s. for \(\pi\)-almost all \(x\in\chi\). Hence if \(\pi(B)> 0\), then \(P_x(M_n\in B_{\text{i.o.}})= 1\) for \(\pi\)-almost all \(x\in\chi\) and it is concluded that every \(\pi\)-positive set \(B\) is recurrent. To a further analysis of IFS, the rate of convergence towards stationarity is given to the highly developed theory for irreducible Harris recurrent Markov chains on general state spaces. Given an IFS of i.i.d. Lipschitz maps satisfying (1) and (2), two questions will be considered. In Section 2 the author considers two equivalent conditions for the positive Harris recurrence on some absorbing subset \(H\) of \(\chi\) (Theorem 2.1) and pr…
Mathematics Subject Classification ID: 60J05
zbMATH DE Number: 1901508
zbMATH Keywords: iterated function system
zbMATH Keywords: Lipschitz map
zbMATH Keywords: Lipschitz constant
zbMATH Keywords: Lyapunov exponent
zbMATH Keywords: Harris recurrence
zbMATH Keywords: total variation
zbMATH Keywords: \(f\)-ergodicity
zbMATH Keywords: geometric ergodicity
zbMATH Keywords: strictly contractive
zbMATH Keywords: drift condition
zbMATH Keywords: level \(\gamma\) ladder epoch
MaRDI profile type: Publication
full work available at URL: a:1022290807360
OpenAlex ID: W2215889408
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description: scientific article; zbMATH DE number 1901508